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adjacent ranges of data arranged in columns or rows. A sample output30

is shown in the next few tables.

Figure 153: Single Factor ANOVA

Table 36: Output from Single Factor ANOVA ” a

ANOVA: Single Factor

Groups Count Sum Average Variance

s1 168 1229.8 7.3 32.7

s2 168 1215.4 7.2 40.1

The first table shows some descriptive statistics for the samples.

Table 37: Output from Single Factor ANOVA ” b

ANOVA

Source of Variation SS Df MS F P“value

I do not supply the sample data for most of the examples in chapter 42 to chapter 46.

30

My experience is that many readers glaze over the examples and do not go through

the difficult step of drawing inferences from a result if the sample data results are

the same as those in the examples in the book.

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Statistical Analysis with Excel

ANOVA

Between Groups 0.62 1 0.62 0.017 0.90

Within Groups 12158.65 334 36.403

Total 12159.27 335

Interpreting the output

The information on “Between Groups” is derived from the difference in

means and variances across the groups. In an ANOVA, the number of

groups may exceed two.

” The test is analyzing the variance as measured by the SS “Sum of

Squares” of the “dependent” series. The total Sum of Squares is

12159.27. Of this, 0.62 can be explained by the differences across the

means of the two groups. The other 12158.65 is explained by the

differences across individual values of the “dependent” series.

• Sum of Squares = Sum of Squares for Between Groups + Sum of

Squares for Within Groups

” The MS is the “Mean Sum of Squares” and is estimated by dividing the

SS by the degrees of freedom. Therefore, the MS for “Between Groups”

equals (0.62/1) = 0.62. (Note that “ANOVA = Analysis of Variance.”)

The MS for “Within Groups” equals (12158.65/334) = 36.403. The MS

may be informally interpreted as “Sum of Squares Explained per Degree

of Freedom.”

• Mean Sum of Squares = (Sum of Squares)/ (Degrees of Freedom)

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Chapter 11: Hypothesis Testing

” The ANOVA uses an F-test to determine if “Between Groups”

information (the number 0.62 in the column “Between Groups” Source

of Variation MS) provides sufficient additional information to improve

the ability of the data to explain the variance in the “dependent” series.

The ANOVA is asking “Does the Between Groups Sum of Squares

Explained per Degree of Freedom” divided by the “Within Groups Sum

of Squares” provide an F that is large enough to justify the statement

“The use of Between Groups information explains a statistically

significant amount of the Sum of Squares of the dependent series.”

• F = (Mean Sum of Squares Between Groups)/ (Mean Sum of

Squares Within Groups)

” All ANOVA tests (including the ANOVA output from a regression) can

be interpreted in the same way “

• F = [ (Increase in ability of model to explain the Sum of

Squares)/ (Degrees of Freedom) /

(Total Sum of Squares) / (Degrees of Freedom)]

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Statistical Analysis with Excel

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Statistical Analysis with Excel

CHAPTER 12

REGRESSION